Invariants
| Level: | $24$ | $\SL_2$-level: | $8$ | Newform level: | $64$ | ||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $1 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (none of which are rational) | Cusp widths | $4^{4}\cdot8^{4}$ | Cusp orbits | $2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $0$ | ||||||
| $\Q$-gonality: | $2$ | ||||||
| $\overline{\Q}$-gonality: | $2$ | ||||||
| Rational cusps: | $0$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8F1 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.96.1.1146 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}1&2\\20&11\end{bmatrix}$, $\begin{bmatrix}3&10\\8&21\end{bmatrix}$, $\begin{bmatrix}11&20\\16&17\end{bmatrix}$, $\begin{bmatrix}23&10\\0&17\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | Group 768.1089047 |
| Contains $-I$: | no $\quad$ (see 24.48.1.u.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $8$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $768$ |
Jacobian
| Conductor: | $2^{6}$ |
| Simple: | yes |
| Squarefree: | yes |
| Decomposition: | $1$ |
| Newforms: | 64.2.a.a |
Models
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 3 x y - 3 y^{2} + 2 w^{2} $ |
| $=$ | $3 x^{2} - 3 x y - 3 y^{2} - z^{2} + 2 w^{2}$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 36 x^{4} + 3 x^{2} y^{2} - z^{4} $ |
Rational points
This modular curve has real points and $\Q_p$ points for $p$ not dividing the level, but no known rational points.
Maps to other modular curves
$j$-invariant map of degree 48 from the embedded model of this modular curve to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle 2^4\,\frac{(z^{2}-2zw+2w^{2})^{3}(z^{2}+2zw+2w^{2})^{3}}{w^{8}z^{4}}$ |
Map of degree 1 from the embedded model of this modular curve to the plane model of the modular curve 24.48.1.u.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle 2z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle 2w$ |
Equation of the image curve:
| $0$ | $=$ | $ 36X^{4}+3X^{2}Y^{2}-Z^{4} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 8.48.1-8.d.1.4 | $8$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.1-8.d.1.6 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.48.0-12.c.1.8 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-12.c.1.9 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.l.1.1 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
| 24.48.0-24.l.1.15 | $24$ | $2$ | $2$ | $0$ | $0$ | full Jacobian |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.192.1-24.ba.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.ba.2.5 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bc.1.2 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bc.2.7 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.be.1.1 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.be.2.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.1.3 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.192.1-24.bg.2.4 | $24$ | $2$ | $2$ | $1$ | $0$ | dimension zero |
| 24.288.9-24.dp.1.20 | $24$ | $3$ | $3$ | $9$ | $1$ | $1^{8}$ |
| 24.384.9-24.ca.1.6 | $24$ | $4$ | $4$ | $9$ | $2$ | $1^{8}$ |
| 120.192.1-120.dc.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dc.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.de.1.12 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.de.2.10 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dg.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.dg.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.di.1.14 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.192.1-120.di.2.2 | $120$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 120.480.17-120.bn.1.8 | $120$ | $5$ | $5$ | $17$ | $?$ | not computed |
| 168.192.1-168.dc.1.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dc.2.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.de.1.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.de.2.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dg.1.3 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.dg.2.5 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.di.1.4 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 168.192.1-168.di.2.7 | $168$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dc.1.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dc.2.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.de.1.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.de.2.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dg.1.3 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.dg.2.5 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.di.1.6 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 264.192.1-264.di.2.7 | $264$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dc.1.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dc.2.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.de.1.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.de.2.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dg.1.5 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.dg.2.3 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.di.1.7 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |
| 312.192.1-312.di.2.6 | $312$ | $2$ | $2$ | $1$ | $?$ | dimension zero |